| L(s) = 1 | + 3-s + 5-s + 9-s − 6·13-s + 15-s − 2·17-s − 8·19-s + 8·23-s + 25-s + 27-s + 2·29-s − 4·31-s + 2·37-s − 6·39-s + 6·41-s − 4·43-s + 45-s − 8·47-s − 2·51-s − 10·53-s − 8·57-s + 4·59-s − 2·61-s − 6·65-s − 4·67-s + 8·69-s − 12·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1/3·9-s − 1.66·13-s + 0.258·15-s − 0.485·17-s − 1.83·19-s + 1.66·23-s + 1/5·25-s + 0.192·27-s + 0.371·29-s − 0.718·31-s + 0.328·37-s − 0.960·39-s + 0.937·41-s − 0.609·43-s + 0.149·45-s − 1.16·47-s − 0.280·51-s − 1.37·53-s − 1.05·57-s + 0.520·59-s − 0.256·61-s − 0.744·65-s − 0.488·67-s + 0.963·69-s − 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.032756594\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.032756594\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.70718311756801, −14.24730955726925, −13.47991378519991, −12.99079080671071, −12.71192260134020, −12.22993971638427, −11.35218084537885, −10.98297513527322, −10.32067558484501, −9.906746109707224, −9.264976391969128, −8.908676915254073, −8.369401570558491, −7.624942831429000, −7.216054144267384, −6.564856683731167, −6.159602102024275, −5.197923831999591, −4.726313740864635, −4.331179485630834, −3.349310463175362, −2.764156666495984, −2.192853096675773, −1.627898078241162, −0.4607531177656057,
0.4607531177656057, 1.627898078241162, 2.192853096675773, 2.764156666495984, 3.349310463175362, 4.331179485630834, 4.726313740864635, 5.197923831999591, 6.159602102024275, 6.564856683731167, 7.216054144267384, 7.624942831429000, 8.369401570558491, 8.908676915254073, 9.264976391969128, 9.906746109707224, 10.32067558484501, 10.98297513527322, 11.35218084537885, 12.22993971638427, 12.71192260134020, 12.99079080671071, 13.47991378519991, 14.24730955726925, 14.70718311756801
